Numerical methods for the Stokes and Navier-Stokes equations driven by threshold slip boundary conditions

In this article, we discuss the numerical solution of the Stokes and Navier-Stokes equations completed by nonlinear slip boundary conditions of friction type in two and three dimensions. To solve the Stokes system, we first reduce the related variational inequality into a saddle point-point problem for a well chosen augmented Lagrangian. To solve this saddle point problem we suggest an alternating direction method of multiplier together with finite element approximations. The solution of the Navier-Stokes system combines finite element approximations, time discretization by operator splitting and augmented Lagrangian method. Numerical experiment results for two and three dimensional flow confirm the interest of these approaches.